Twisted Bhargava Cubes

TL;DR

This paper explores the orbit structure of a specific algebraic action related to the triality D_4 group, extending classical results on Bhargava cubes to a more complex setting over arbitrary fields.

Contribution

It introduces a new perspective on the orbit structure of the triality D_4 case, generalizing known results from algebraically closed fields to broader contexts.

Findings

Orbit structure characterized for triality D_4

Identification of invariants in non-algebraically closed fields

Extension of Bhargava cube analysis to new algebraic settings

Abstract

Let G be a reductive group and P=MN a maximal parabolic subgroup. The group M acts, by conjugation, on N/[N,N]. It is well known that, over an algebraically closed field, the group M acts transitively on a Zariski open set. However, over a general field, the structure of orbits may be quite non-trivial. A description may involve unexpected invariants. A notable example is when G is a split, simply connected group of type D_4, and P is the maximal parabolic corresponding to the branching point of the Dynkin diagram. The space N/[N,N] is also known as the Bhargava cube, and it was the starting point of his investigations of prehomogeneous spaces. We consider a version of this problem for the triality D_4.